The quench induced dynamical topological phase has been mainly studied in Hermitian systems, while the system is generally non-Hermitian when coupled to environment [1]}. Recently, the interplay between non-Hermiticity and topology has attracted considerable attention [2]}, [3]}, with rich phenomena being uncovered, such as the exotic topological phases driven by exceptional points [4]}, [5]}, [6]}, [7]}, the anomalous bulk-boundary correspondence [5]}, [9]}, [10]}, [11]}, and the non-Hermitian skin effect [12]}. Experimental observations of the non-Hermitian topological physics have been reported in classical systems with gain and loss, like the photonic systems [13]}, [14]}, the active mechanical metamaterial [15]}, as well as topolectrical circuits [16]}, and in quantum simulators, like the nitrogen-vacancy center [17]}, [18]}, where the non-Hermitian effects are engineered by coupling to auxiliary qubits.
